Deviation bounds for additive functionals of Markov process

Abstract

In this paper we derive non asymptotic deviation bounds for $$¶_ν(|\frac 1t \int_0^t V(X_s) ds - \int V dμ| \geq R)$$ where $X$ is a $μ$ stationary and ergodic Markov process and $V$ is some $μ$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $μ$. Regularity means here that $μ$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc...).